Something like a lavender needs a very light banana peel like color or fully white to take. Your base still looks a bit too grey imo for the lavender to have taken correctly. I think you might be better off taking a higher pigmented purple like SFX deep purple and diluting it w white conditioner to take over better what your current color is.

"If you don't know where you're going, any road will get you there." <---Lewis Carrol

You said your hair is 'natural white', is that from greying of your hair? , because naturally grey hair can be very dye resistant, it's hard to get any dye to stick. Trying to soften it first with a peroxide-only soak has helped other people. Their are some older forum posts on it.